一类刻画微机电模型四阶抛物型方程解的适定性

数学物理学报 ›› 2021, Vol. 41 ›› Issue (6): 1718-1733.

一类刻画微机电模型四阶抛物型方程解的适定性

赖柏顺(),罗勍*()

^{湖南师范大学数学与统计学院长沙 410081}

收稿日期:2020-09-18 出版日期:2021-12-26 发布日期:2021-12-02
通讯作者: 罗勍 E-mail:laibaishun@hunnu.edu.cn;10100095@vip.henu.edu.cn
作者简介:赖柏顺, E-mail: laibaishun@hunnu.edu.cn
基金资助:
国家自然科学基金(11971148)

Well-Posedness of a Fourth Order Parabolic Equation Modeling MEMS

Baishun Lai(),Qing Luo*()

^{School of Mathematics and Statistics, Hunan Normal University, Changsha 410081}

Received:2020-09-18 Online:2021-12-26 Published:2021-12-02
Contact: Qing Luo E-mail:laibaishun@hunnu.edu.cn;10100095@vip.henu.edu.cn
Supported by:
the NSFC(11971148)

摘要/Abstract

摘要：

该文考虑了有界区域上一类带奇异非线性项$（1-u）^{-2}$的四阶演化方程解的适定性.该模型具有很强的物理背景，它刻画了微机电模型的工作原理.首先，该文研究了四阶抛物方程解的适定性，该方程的适定性在文章[1]中通过半群理论得到验证，但是他们只限于区域空间维数为2维的情形.通过Faedo-Galerkin技巧，可以证明该方程在空间维数小于等于7的情形下解是适定的.

关键词: 微机电模型, 四阶演化方程, 适定性

Abstract:

In this paper, we consider a fourth order evolution equation involving a singular nonlinear term $\frac{\lambda}{(1-u)^{2}}$ in a bounded domain $\Omega \subset \mathbb{R}^{n}$. This equation arises in the modeling of microelectromechanical systems. We first investigate the well-posedness of a fourth order parabolic equation which has been studied in [1], where the authors, by the semigroup argument, obtained the well-posedness of this equation for $n\leq2$. Instead of semigroup method, we use the Faedo-Galerkin technique to construct a unique solution of the fourth order parabolic equation for $n\leq7$, which completes the result of [1].

Key words: Electrostatic MEMS, Fourth order evolution equation, Well-posedness

中图分类号:

O175

赖柏顺,罗勍. 一类刻画微机电模型四阶抛物型方程解的适定性[J]. 数学物理学报, 2021, 41(6): 1718-1733.

Baishun Lai,Qing Luo. Well-Posedness of a Fourth Order Parabolic Equation Modeling MEMS[J]. Acta mathematica scientia,Series A, 2021, 41(6): 1718-1733.

参考文献 26

1	Laurencot P , Walker C . A fourth-order model for MEMS with clamped boundary conditions. Proc London Math Soc, 2014, 3, 1435- 1464
2	Pelesko J A , Bernstein A A . Modeling MEMS and NEMS. London: Chapman Hall & CRC Press, 2002
3	Taylor G I . The coalescence of closely spaced drops when they are at different electric potentials. Proc Royal Soc A, 1968, 306, 423- 434
4	Nathanson H C , Newell W E , Wickstrom R A , Davis J R . The resonant gate transistor. IEEE Transactions on Electron Devices, 1967, 14, 117- 133 doi: 10.1109/T-ED.1967.15912
5	Guo Y , Pan Z , Ward M J . Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties. Siam J Appl Math, 2005, 66, 309- 338 doi: 10.1137/040613391
6	Lin F , Yang Y . Nonlinear non-local elliptic equation modelling electrostatic actuation. Proc Roy Soc Lond Ser A Math Phys Eng Sci, 2007, 463, 1323- 1337
7	Escher J , Laurencot P , Walker C . A parabolic free boundary problem modeling electrostatic MEMS. Arch Ration Mech Anal, 2004, 211, 389- 417
8	Cowan C , Esposito P , Ghoussoub N , Moradifam A . The critical dimension for a forth order elliptic problem with singular nonlineartiy. Arch Ration Mech Anal, 2010, 198, 763- 787 doi: 10.1007/s00205-010-0367-x
9	Guo Z , Wei J . On a fourth order nonlinear elliptic equation with negative exponent. SIAM J Math Anal, 2009, 40, 2034- 2054 doi: 10.1137/070703375
10	Lai B S , Ye D . Remarks on entire solutions for two fourth order elliptic problems. Proc Edinb Math Soc, 2016, 59, 777- 786 doi: 10.1017/S0013091515000371
11	Lai B S . Regularity and stability of solutions to semilinear fourth order elliptic problems with negative exponents. Proc Roy Soc Edinburgh Sect A, 2016, 146, 195- 212 doi: 10.1017/S0308210515000426
12	Guo Y . Dynamical solutions of singular wave equations modeling electrostatic MEMS. SIAM J Appl Dyn Syst, 2010, 9, 1135- 1163 doi: 10.1137/09077117X
13	Esposito P , Ghoussoub N , Guo Y . Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS. Providence, RI: American Mathematical Society, 2010
14	Guo J S , Hu B , Wang C J . A nonlocal quenching problem arising in a micro-electro mechanical system. Quart Appl Math, 2009, 67, 725- 734 doi: 10.1090/S0033-569X-09-01159-5
15	Lindsay A E , Lega J . Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a MEMS capacitor. Siam J Appl Math, 2012, 72, 935- 958 doi: 10.1137/110832550
16	Lindsay A E , Lega J , Sayas F J . The quenching set of a MEMS capacitor in two-dimensional geometries. J Nonlinear Sci, 2013, 23, 807- 834 doi: 10.1007/s00332-013-9169-2
17	Gazzola F , Grunau H C , Sweers G . Polyharmonic Boundary Value Problems. Berlin: Springer, 2010
18	Evans L C . Partial Differential Equations. Providence, RI: American Mathematical Society, 1998
19	Laurencot P , Walker C . A stationary free boundary problem modeling electrostatic MEMS. Arch Ration Mech Anal, 2013, 207, 139- 158 doi: 10.1007/s00205-012-0559-7
20	Teman R . Navier-Stokes Equations: Theory and Numerical Analysis. Amsterdam: North-Holland, 1977
21	Caldiroli P , Musina R . Rellich inequalities with weights. Calc Var PDE, 2012, 45, 147- 164 doi: 10.1007/s00526-011-0454-3
22	Guo Z , Lai B S , Ye D . Revisit the biharmonic equation modelling electrostatic actuation in lower dimensions. Proceeding Amer Math Soc, 2014, 142, 2027- 2034 doi: 10.1090/S0002-9939-2014-11895-8
23	Kavallaris N I , Lacey A A , Nikolopoulos C V , Tzanetis D E . A hyperbolic non-local problem modelling MEMS technology. Rocky Mt J Math, 2011, 41, 505- 534
24	Liang C C , Li J Y , Zhang K J . On a hyperbolic equation arising in electrostatic MEMS. J Differential Equations, 2014, 256, 503- 530 doi: 10.1016/j.jde.2013.09.010
25	Chang P H , Levine H A . The quenching of solutions of semilinear hyperbolic equations. Siam J Math Anal, 1981, 12, 893- 903 doi: 10.1137/0512075
26	Lacey A A . Mathematical analysis of thermal runaway for spatially inhomogeneous reactions. SIAM J Appl Math, 1983, 43, 1350- 1366 doi: 10.1137/0143090

[1]	闵建中,刘宪高,刘子轩. 不可压液晶方程组的Serrin解[J]. 数学物理学报, 2021, 41(6): 1671-1683.
[2]	孙丹丹, 李盈科, 滕志东, 张太雷. 具有年龄结构的麻疹传染病模型的稳定性分析[J]. 数学物理学报, 2021, 41(6): 1950-1968.
[3]	西宣宣,侯咪咪,周先锋. 非自治Caputo分数阶发展方程弱解的适定性与不变集[J]. 数学物理学报, 2021, 41(1): 149-165.
[4]	蔡钢. Banach空间上有限时滞退化微分方程的适定性[J]. 数学物理学报, 2018, 38(2): 264-275.
[5]	Marko Kostić, 李成刚, 李淼. 抽象多项Riemann-Liouville分数阶微分方程[J]. 数学物理学报, 2016, 36(4): 601-622.
[6]	曲风龙, 李希亮. 各向同性的麦克斯韦方程的内部传输问题及反射参数的估计[J]. 数学物理学报, 2013, 33(6): 1178-1188.
[7]	朱莉, 夏福全. 广义混合变分不等式的Levitin-Polyak适定性[J]. 数学物理学报, 2012, 32(4): 633-643.
[8]	尹姝馨; 许跟起. 具有多重严重故障和非严重故障和修复功能的系统的可靠性分析[J]. 数学物理学报, 2007, 27(3): 392-413.
[9]	倪仁兴. 广义共同逼近问题的适定性[J]. 数学物理学报, 2003, 23(2): 161-168.
[10]	郭定辉. 一类广义层流方程组的初值问题的局部适定性[J]. 数学物理学报, 2001, 21(4): 433-438.
[11]	刘康生. 非均质储层储量再估计的适定性[J]. 数学物理学报, 1996, 16(S1): 42-47.
[12]	郑家茂, 王元明. 用弹性波方程反演层状地球模型中的ρ,λ,μ[J]. 数学物理学报, 1993, 13(3): 280-290.
[13]	高西玲. 关于一类椭圆型方程的衔接问题[J]. 数学物理学报, 1993, 13(3): 353-360.